Reassessing chain tilt in the lamellar crystals of polyethylene

Semicrystalline polymers are extensively used in various forms, including fibres, films, and bottles. They exhibit remarkable properties, e.g., mechanical and thermal, that are governed by hierarchical structures comprising 10–20-nm-thick lamellar crystals. In 1957, Keller deduced that long polyethylene (PE) chains fold to form thin single lamellar crystals, with the molecular chains perpendicular to the flat faces of the crystals (the chain-folding model). Chains inclining to the perpendicular orientation in single crystals have since been reported, along with their effects on the physical properties of PE. For bulk specimens, the chain tilt angle (φ) has been investigated only for model samples with well-annealed internal structures. However, for briefly annealed specimens, the φ values of lamellae and their origins are controversial owing to the disordered lamellar morphology and orientation. Herein, we report the direct determination of molecular-chain orientations in the lamellar crystals of high-density PE using a state-of-the-art electron-diffraction-based imaging technique with nanometre-scale positional resolution and provide compelling evidence for the existence of lamellar crystals with different inner-chain orientations. Clarifying the nanoscale variation in lamellar crystals in PE can allow precise tuning of properties and expedite resource-saving material design.

diffraction (ED).Supplementary Fig. 3a shows the WAXD pattern of PE, in which the diffraction rings of {hk0}-including the 110, 200, and 020 reflections of orthorhombic PE-are clearly observed.This result unambiguously indicates the random orientation of crystallites within the irradiation area (2 × 10 5 μm 2 ).The crystallinity of PE was calculated to be 58.9% using the peak separation method.Supplementary Fig. 3b shows the SAXS pattern of PE.A ring pattern implies a random orientation of lamellar stacks comprising the lamellar crystals and amorphous layers within the irradiation area (1 × 10 5 μm 2 ).The long period (Lp), lamellar thickness (lc), and amorphous layer thickness (la) were calculated to be 30.3,17.8, and 12.5 nm, respectively, where Lp = lc + la.Notably, the WAXD crystallinity exceeded 50%; hence, longer and shorter spacings were assigned to lc and la, respectively, in the SAXS correlation function analysis.Fig. 2h shows the microbeam ED pattern of PE acquired under moderate conditions [sample temperature: -175 ℃, dose rate: 0.1 e -/(Å 2 s), total dose: 0.25 e -/Å 2 ].Although the electron-beam area on the sample (~3 μm 2 ) was several tens of thousands of times smaller than the X-ray beam area (1-2 × 10 5 μm 2 ), the ED exhibited a ring-like pattern similar to the WAXD pattern.
Although these methods provided statistically averaged information, local structural information was lost by averaging.

Image reconstruction
In this analysis, information on the diffraction disc intensity and azimuth was extracted by performing image processing on each ED pattern.For more detailed analysis of the crystal orientation and overlapping of multiple crystallites, we plan to incorporate methods such as clustering and pattern matching in the future.
1.A circularly averaged profile (blue line in Supplementary Fig. 5a) was created for the average pattern of all ED patterns acquired via nanodiffraction imaging (NDI).The intensity of the profile decreased with an increase in the scattering angle, and diffraction peaks appeared at approximately 1.6-2.7 nm -1 .A background (BG) function was created by fitting two exponential functions [f(q) = ae -q/b ] to the region excluding these peaks (the black dashed line in Supplementary Fig. 5a).
2. A BG pattern (Supplementary Fig. 5b) was created using the BG function setup in 1 and subtracted from the raw ED pattern (Supplementary Fig. 5c) to create the pattern shown in Supplementary Fig. 5d.In the raw ED pattern, the intensities around the centre were almost identical to those of the diffraction disc owing to BG and noise.
In comparison, in the BG-subtracted pattern, the intensity ratio of the diffraction disc to other parts was higher.
3. The diffraction discs observed in the ED pattern often did not have the same intensity of pairs symmetric with respect to the centre and did not necessarily exhibit the diameter expected from the convergence angle.The first reason for this is that the incident electron beam deviated slightly from the ideal incident direction.The second reason is the presence of an intensity distribution inside the diffraction disc.
Considering the second reason, in contrast to X-ray and neutron diffraction, in which single scattering (kinematic diffraction) is almost established, electron diffraction is considerably affected by multiple scattering (dynamical diffraction).Furthermore, the ED pattern obtained by the focused electron beam in this analysis corresponded to observing the diffraction intensity with slightly different incident angles simultaneously.Therefore, the intensity distribution inside the disc due to these effects contained crucial crystallographic information; however, an ED pattern suitable for such detailed analysis could not be obtained in these measurements.
Alternatively, as a process suitable for polymer specimens, we calculated the intensity and azimuth as a pair instead of a single diffraction disk.Supplementary Fig. 5e shows results that are the sum of those in Supplementary Fig. 5d and those rotated 180° with respect to the centre.This process recovered the disc shape, including weak diffraction points, and the error in the azimuth values was reduced to ~1°.
4. Finally, the processes for removing the slight noise and extracting only the diffraction disk were performed.Supplementary Figs.5f and g were created using Gaussian filters with two different standard deviations (σ = 2 and 15 pixels; disc diameter = 10 pixels) from Supplementary Fig. 5e, respectively.Then, Supplementary Fig. 5h was created by determining the ratio of the two sets of results (Supplementary Fig. 5f / Supplementary Fig. 5g).Pseudo-peaks appearing at the outer edges of the image were excluded during the image reconstruction stage.5. Steps 2-4 were performed for all 600 × 600 ED patterns.

Rotation of masks and azimuthal profile creation
A schematic of profile creation is shown in Supplementary Fig. 5i.Azimuth profiles were created from the raw ED pattern (dotted lines in Supplementary Fig. 5j) and the processed pattern (solid lines in Supplementary Fig. 5j).For the raw-pattern profiles, the intensity of the 110-region profile (blue) was always higher than that of the 200-region profile (orange) owing to the inelastic scattering BG.A comparison of two sets of profiles revealed that the processed-pattern profiles had sharper peaks at β = 25°, 87°, and 155°.This allowed automatic determination of the peak top and calculation of the azimuth even for spots with low intensities.Furthermore, the intensity of the blue profile was higher than that of the orange profile at β = 25° and 87°, with the opposite trend occurring at β = 155°.Thus, the aforementioned process separated the 110 and 200 reflection intensities through a hexagonal-like pattern.
For each scanning position (x, y), azimuthal profiles were created as described in the previous section.The maximum intensities of the 200 reflection profiles were calculated and plotted to obtain the reconstructed dark-field scanning TEM (DF-STEM) image shown in Fig. 3a.

Relationship between electron-beam direction and c-axis
The relationship between the electron-beam direction and c-axis was established using the following four ED pattern rules: When a hexagonal-like ED pattern was obtained (for example, Fig. 2e), the electron beam was parallel to the c-axis.(The c-axis was oriented perpendicular to the plane of the image.) (ii) When 002 spots were observed, the electron beam was perpendicular to the c-axis.
(iii) When two hk0 spots were observed, the angle between the electron beam and the c-axis (molecular chain) could not be determined.
(iv) When the flat faces of the lamellar crystals were {h0l} planes [1][2][3][4] , the lamellae viewed along the b-axis appeared to be edge-on.In this case, the c-axis was perpendicular to the electron beam (and parallel to the plane of the image), and a pair of 200 spots were observed in the ED patterns (Supplementary Fig. 8).
When the lamellar crystals were viewed edge-on, an image of the lamellae with clear contrast was acquired.However, as the lamellae inclined away from the observing (electron-beam) direction, the apparent lamellar thickness (lc') increased, and the contrast became ambiguous.The relationship between the tilting angle of the lamellae (ψ) and lc' is shown in Supplementary Fig. 7, which was constructed using the lamellar thickness estimated via SAXS (lc = 18 nm).When ψ reached 7°, lc' was equal to the long spacing value (30.3 nm; lc + amorphous layer thickness), and the lamellar and amorphous domains merged into a single region.Therefore, when the ~18-nm-thick lamellae were visualised edge-on (revealing clear boundaries between the lamellar and amorphous domains), the tilting angles of the lamellae against the electron-beam direction were <7°.Additionally, when the lamellae reconstructed from the 200 spots were visualised edge-on, the electron beam was almost perpendicular to the c-axis (90 ± ~7°).

Evaluating distribution of tilting angles
1.The lamellar crystals that could be separated from the amorphous and adjacent lamellar crystals were selected from the reconstructed image, and their azimuth angles were measured as straight lines.The lamellar crystals with bends or cranks were divided at the bends to make them straight.
2. The ED pattern of the target area in the lamellar crystal was extracted, and an azimuth profile was created by rotating the circular masks at the scattering angle of the 200 reflection (the pair of orange masks shown in Supplementary Fig. 5i).The azimuth exhibiting the maximum intensity in this profile (orange solid line in Supplementary Fig. 5j) was treated as that of the diffraction spot.
3. Finally, the difference between the azimuth angle (of the normal) of the lamellar crystal and that of the spot (the chain oriented along the normal to the line connecting the spots) in the ED pattern was calculated and used as a histogram.
Notably, dozens of ED spot angles existed for each azimuth angle of a part of the lamellar crystal.The results for multiple lamellar crystals were summarised in one histogram.
Two types of errors could occur in the measurement of the azimuth angles: (1) the error in measuring the azimuth angles of lamellar crystals and (2) that in measuring the azimuth angles of ED spots automatically.The results were eliminated when the error was (1)     several degrees and (2) significantly large.However, not all ED patterns were checked; therefore, the deviation could have been >5° in certain cases.
Two pairs of circular masks that were symmetric about the centre were created to include the 110 (blue) and 200 (orange) spots and azimuthally scanned.The azimuth angle (β) is defined according to the right-horizontal direction with an anticlockwise rotation being the origin.j, Azimuth profiles created by calculating the total intensity inside the two masks per pair at each β.